In this article, related to the calculation of internal forces, we will deal with the determination of axial forces in truss members. The main topic of this entry is Ritter’s method.
In this article, we will cover the following subjects:
Internal forces in planar panels
After reading the article on the method of joints, you already know that the axial forces in truss members are constant (independent of the location along a given member). You also know that the most straightforward method of analyzing these forces is to consider the equilibrium of forces at each truss node. This is a fairly simple but unfortunately time-consuming procedure, so we mainly use it when we need to determine the forces in all the truss members. The truth is that when designing a real truss, we indeed have to calculate the forces in all members anyway—since each rod must have its cross-section properly designed.
Sometimes, however (mainly due to the wishes of examiners 😊), a problem requires us to calculate the internal forces only in a specific group of members. Most often, these members are located in the very center of the truss, which means that using the method of joints to “reach” them would take far too much time. And this is where Ritter’s method comes to our aid!
In Ritter’s method, we cut through the truss along a chosen set of members, which allows us to analyze its local equilibrium—almost in the same way as when analyzing internal forces in beams or frames. The key difference is that at the cut we do not obtain a single set of axial force, shear force, and bending moment. Instead, the normal forces of the cut members appear at the section. Such a local system—just like in the case of frames or beams—must satisfy the equilibrium equations in order to remain at rest. In other words, we can use the well-known static equilibrium equations to determine the forces in the truss members.
Ritter’s method enables the calculation of member forces in a truss based on the equilibrium of a locally cut-out part of the truss.
Application of Ritter’s method
Let us now think for a moment about the actual application of Ritter’s method. In the general case, the procedure is very simple: we take the truss, cut it through some members, replace them with axial forces, write the equilibrium equations for one of the cut-out parts of the truss (including, of course, only the loads and reactions associated with the analyzed part of the truss), calculate the forces in the members—and voilà. Things become more complicated, however, when we want to apply this method—first—correctly, and second—efficiently 😉.
Regarding the word correctly—there is a main rule that limits the possibility of using Ritter’s method in the context of analyzing forces in truss members. There are also several myths associated with it (not to mention half-truths or outright misconceptions…) which are regularly repeated by instructors and which I often hear from students 😉. I will now describe all of them:
Myth 1. Ritter’s cut must be made along a straight line.
This is not true—you can cut through successive members by passing between adjacent elementary cells. I wrote in detail about what elementary cells in trusses are in the first article on support reaction calculations—I highly recommend checking it out 😊.
Myth 2. Ritter’s cut MUST pass through 3 members…
It doesn’t have to… Which does not mean that cutting through more members will always give us something useful in terms of calculations. I think this myth deserves a broader comment. The matter is not really about mechanics, but about mathematics!
If we cut through three truss members (assuming we have already calculated the support reactions), we end up with three unknowns in the system of equations. At our disposal, we also have three static equilibrium equations (sum of forces in x, sum of forces in y, and sum of moments). Such a system (if Rule 1 is satisfied) allows us to directly calculate the values in the three cut members.
If we cut through more members, then the system cannot be solved completely—we might be able to determine one or two forces from the equilibrium equations, but we definitely won’t be able to solve for all of them… unless… we already know the values of the forces in some of the cut members!!! And this is the main flaw in this myth—if we already know the forces in certain members from previous calculations, then we can cut through them freely without any consequence for Ritter’s method! In other words—we always look at it from a mathematical perspective: we can have a maximum of 3 unknowns with 3 equilibrium equations—regardless of how many members we cut. However, to be sure that we will calculate the axial forces, we must stick to Rule 1.
Rule 1. To be able to calculate the values of the forces in three cut members, they must not all intersect at a single point, and they must not all be parallel to one another.
In such cases, we would either be dealing with a concurrent force system (no possibility of using moment equilibrium), or with a system where the equilibrium equation in the direction perpendicular to the members would contain no unknowns. Either way—we would not be able to determine the values in all the members.
That’s all as far as the assumptions of Ritter’s method are concerned. Let us now move on to the word “efficiently.” Here’s a handful of tips for students so that solving with this method does not generate more problems than necessary 😉.
Suggestion 1 – the side of the cut.
This is probably the most important issue when it comes to the speed of solving. Regardless of which side we choose, the system of unknowns in the form of axial forces is always the same, so the configuration of the equilibrium equations will not change. What usually changes, however, is the number of loads and reactions that we must take into account in the equilibrium equations. The more forces are present on a given part, the more terms we will need to write in the equilibrium equations. Hence—the conclusion is simple—it is best to choose the side where there are as few additional forces as possible. This minimizes the number of sign mistakes, calculation errors, and ultimately improves the speed of obtaining the result!
Suggestion 2 – the set of equilibrium equations.
As you surely know—or if you’ve read even a few articles on the MechaDevs BLOG you’ve already learned—in planar systems we almost always write three fundamental equilibrium equations:
∑F_{ix} =0,∑F_{iy} =0,∑M_{iz} =0They allow us to calculate support reactions, internal forces, and so on. In the case of Ritter’s method, as you also know, they allow us to determine the axial forces in the members… unfortunately, in most cases, this system of three equations leads to a coupled system of equations. This means that in at least one equation there are two unknowns, so in order to calculate everything, we have to transform and substitute between the equations, and so on. And now I might surprise you! It’s not the case that the system can only be solved with the three standard equilibrium equations! In practice, we can just as well write three moment equations—provided that the points with respect to which we take the moments are not collinear 😊.
If we relate this to the situation in Ritter’s method and Rule 1 is satisfied, then very often (though not always) each pair of our three cut members intersects at a different point. This means that if we write moment equations with respect to those points, we get three independent equations, each containing only a single unknown—no substitutions or extra manipulations required. We just isolate the unknown in the equation and we’re done 😉. The situation where the above description does not apply occurs when two of the three cut members are parallel. Their values can be calculated using two moment equations, while the value in the third member is found using the force equilibrium in the direction perpendicular to those two members 😉.Either way—we end up with a set of equations that practically does not require any transformations—take a look at the figure below!
To summarize—Ritter’s method is an interesting way of determining the axial forces in truss members, and with the use of certain tricks it allows for very quick calculation of the force in a specific member. For this reason, it is most often used precisely when determining the forces in selected groups of members!
If you’ve made it this far, thank you for reading the article, and I encourage you to try out the EquiTruss module! 😊 I also invite you to check out the article on the analysis of zero-force members!
